DETERMINING THE VALUE OF DOUBLE BARRIER OPTION USING STANDARD MONTE CARLO, ANTITHETIC VARIATE, AND CONTROL VARIATE METHODS

Article History: In this paper, we applied the standard Monte Carlo, antithetic variate


INTRODUCTION
The development of financial assets in Indonesia today is increasing rapidly. Referring to the data from Kustodian Sentral Efek Indonesia (KSEI), by December 2021, stock market investors have increased to 92,99% in 2020 to 7,4 million investors [1]. The most popular investment product in stock is because it gives a higher yield. The risk in stock investments is also high, so it is necessary to hedge. As an alternative to hedge the risk, investors can use derivative products, one of which is an option.
According to Hull, exotic options are much more profitable than plain vanilla options [2]. One of the widely used exotic options is the barrier option because it has a barrier that gives additional protection for the investors and allows them to participate in determining the barrier. On the barrier option, payoff and option value depend on the achievement of a barrier from the underlying assets price during a certain period of time [3].
In this paper, we will discuss determining the option value of the double barrier. A double barrier is a barrier option that has two barriers (lower and upper barrier). Double barrier option that has been researched in financial mathematic literature were mostly option of double barrier knock-out as written by [4], [5], [6], [7], [8] and [9]. In addition to the literatures, this paper is focuses in the determination of double barrier knock-in option price. The benefit of double barrier knock-in option compared to the other barrier option is that this option gives protection to the investor when the underlying asset price is highly fluctuated.
Option valuation in the stock market is an important matter used to count the fair value of the options and to avoid the possibility of arbitrage. Black Scholes Model to determine stock price movement becomes fundamental in the calculation of the analytic value of the options. However, the analytic solution of exotic options in general is hard to determine, so numeric methods are needed to value this kind of option.
Various approaches of numeric methods to stipulate the barrier option value have been developed, such as the Monte Carlo method and the binomial lattice method. Noury and Abbasi researched the determination of the option value of double barrier knock-out using a modification of the Monte Carlo method [10]. The idea of this modified method is to use uniformly distributed random numbers and an exit probability of performing a robust estimation of the first time the stock price hits the barrier, resulting in the numeric solution being closer to the analytical value and smaller error when underlying asset touched the barrier for the first time.
In their research, Wang and Wang stated that Monte Carlo simulation is proven to be effective and simple in approximating the option value of barrier down-and-out option [11]. However, Monte Carlo's convergence rate is low, and it needs many simulations to gain a closer result to the analytic solution. The efficiency and accuracy of option value estimation of the barrier option can be increased by using the variance reduction technique. The Monte Carlo method with variance reduction consists of several types of techniques, such as control variates, antithetic variates, stratified sampling, Latin hypercube sampling, matching underlying assets, and importance sampling [12].
Monte Carlo Method of antithetic variates on research by Putri in determining the option value of barrier down-and-out, yielding a smaller error value and also shorter confidence interval compared to those using standard Monte Carlo simulation [13]. This finding is strengthened by Alzubaidi, which in his research, stated that the antithetic variates technique on the option calculation of barrier down-and-out option accelerated the Monte Carlo simulation and gave twice the accuracy than the standard Monte Carlo simulation [14]. Maulida, in her study of using Monte Carlo control variates, gave better calculation results on option value double barrier knock-out and standard deviation, also a smaller error than that of the standard Monte Carlo method [15].
To determine the best method in approximating the value of double barrier knock-in option, several methods were compared in this paper. The methods that were being compared are standard Monte Carlo, antithetic variate, and control variate methods. The values that were being compared are the option values, the relative errors, and the computation time of each method.

RESEARCH METHODS
Steps used in this study are as follow: a. To retrieve secondary data of several stock prices from www.finance.yahoo.com. b. To calculate the return of the retrieved stock price then examined the normality of each return stock. Next, choosing the normal distributed stock, the study used the stock price of PT Aneka Tambang Tbk (ANTM) from April 1, 2019 until March 1, 2022. c. To approximate the parameter value needed in simulating the option prices, which is the risk-free rate ( ), volatility (σ), initial stock price ( 0 ), the strike price ( ), the number of simulations ( ), maturity date ( ), time-span (∆t), upper barrier ( ), lower barrier ( ) and the number of monitoring points ( ).

The Algorithm of Determining the Value of Double Barrier Option using Standard Monte Carlo Method
Steps of determining the value of double barrier option using Standard Monte Carlo method are as follow: a. To generate the random variable of × = ( )~(0,1) with = 1, 2, ⋯ , and = 1, 2, ⋯ , . b. To calculate stock price estimation on every monitoring point using the random variable that has been generated using Equation (1).
c. To approximate the payoff of option call using Equation (2) and put using Equation (3).
d. To calculate call option value using Equation (4) and put using Equation (5).
e. To determine the average of call option value by Equation (6) and put using Equation (7) so it obtained the double barrier option value estimator using standard Monte Carlo method.

The Algorithm of Determining Double Barrier Option Value using Antithetic Variate Method
Steps of determining the double barrier option value using Monte Carlo antithetic variate method are as follow: a. To generate the random variable of × = ( )~(0,1) and − × = (− )~(0,1). b. To calculate stock price estimation on every monitoring point using the Equation (8) and Equation (9).
c. To determine payoff of option call of two stock price estimations using Equation (2) and calculating the average of payoff by Equation (10).
d. To determine the payoff of option put of two stock price estimations by Equation (3) and calculating the average of payoff using Equation (11).
e. To determine the average of call and put options values using Equation (12) and Equation (13) obtaining the double barrier option value estimator using Monte Carlo Antithetic variate method.

The Algorithm of Determining the Double Barrier option value using Control Variate Method
Steps in determining the double barrier option value using Monte Carlo control variate method are as follow [16]: a. To calculate the European call and put options value of Black-Scholes Model using Equation (14) and Equation (15). Then determining the European payoff of option call and put by the same stock price movement using Equation (16) and Equation (17).

RESULTS AND DISCUSSION
Underlying assets used in this paper are the monthly non-dividend stock prices of PT Aneka Tambang   In Figure 1, it can be seen that the stock prices were highly fluctuated, resulting in the investors coming up with strategies it would have high returns with low risks. One of the ways to hedge is using a double barrier knock-in option. The parameter used in computing the option value of double barrier knock-in is as follows.

The Numeric Result of Call Double Barrier Knock-in Option Value
n the calculation of the call double barrier option value using the Monte Carlo control variate method, it was first determined the analytic value from the European call option, which is its control variate. The analytic value of the European call option is the European call option of the Black-Scholes Model, which its value obtained with the help of software RMFI v1.00a amounted of c = 483.94982. The numerical result of call double barrier knock-in option value using standard Monte Carlo, antithetic variate, and control variate methods are presented in Table 1.   Table 1, it could be seen that as the simulation number increases, then the relative error value decreases. The relative error value showed that the control variates method is the fastest method in approaching the exact solution compared to the other methods, while using the standard Monte Carlo method took a long time to converge to the exact solution.

The Numeric Result of Put Double Barrier Knock-in Option Value
In the calculation of the put double barrier option value using the Monte Carlo control variate method, the analytic value was determined first from the European put option, which is its control variate. The European put analytic option value is the European put option of the Black-Scholes Model that its value earned with the help of software RMFI v1.00a amounted of = 602,24901. he numerical result of putting double barrier knock-in option value using the standard Monte Carlo, antithetic variate, and control variate methods were presented in Table 2. The exact value of the double barrier knock-in option is the maximal simulation iteration that could be processed by the available computer, i. e. with M = 15000000 for the calculation of the exact solution. In that simulation, it generated the put option value of the standard Monte Carlo (MC_S) = 601,10478, the call option value of the antithetic variate (MC_AV) = 598,2193, and the call option value of the control variate (MC_CV) = 601,1771. Based on the relative error value in Table 2, it could be seen that as the simulation number increased, then the relative error value decreased. The relative error value showed the control variate method is the fastest in approaching the exact solution compared to the other methods.
In Table 2, it can be seen that the obtained put option value is getting smaller with more simulations done. Different from the call option value, that got bigger with the increasing number of simulations done ( Table 1). The relative error value generated in the standard Monte Carlo and antithetic variate in the calculation of the put option is smaller compared with the calculation of the call option.

Computation Time of the Double Barrier Knock-in Option Value Simulation
The efficiency of the standard Monte Carlo, antithetic variate, and control variate methods can be seen from the time needed to compute the double barrier knock-in option value. The computation time in the simulation of double barrier knock-in option value using the standard Monte Carlo, antithetic variate, and control variate methods were presented in Table 3. Based on Table 3, it could be seen that the more simulations being done, then the time it took to compute the double barrier knock-in option value is also longer. The Standard Monte Carlo method has the shortest computation time compared to any other method. The control variate method has a relatively short computation time compared to the antithetic variate method. The computation time needed for the antithetic variate method is the longest of the other methods.

The Effect of Parameter Change to Double Barrier Knock-in Option Value
Next to be simulated is the random variable estimator of the double barrier knock-in option value of standard Monte Carlo, antithetic variate, and control variate methods with the change of value parameter of the strike price ( ), initial stock price ( 0 ), maturity date ( ), and volatility ( ) by simulation = 100000.  In Figure 2, the call double barrier knock-in option value shown is getting lower as the strike price is gets higher. On the other hand, the put double barrier knock-in option value is getting higher as the strike price is getting higher too. The put double barrier knock-in option value resulted from standard Monte Carlo, antithetic variate and control variate methods are relatively the same, while in the calculation of call of standard, Monte Carlo earned a little different option value than the other methods. b.
The Effect of Initial Stock price ( 0 ) to the option value The effect of the Initial stock price changes to the double barrier knock-in option value by standard Monte Carlo, antithetic variate, and control variate methods presented in Figure 3. In Figure 3, the call double barrier knock-in option value is seen to gets bigger as the Initial stock price get bigger too. On the contrary, the put double barrier knock-in option value is getting lower as the Initial stock price gets higher. The option value generated from the antithetic variate and control variate methods has relatively the same values. However, the standard Monte Carlo method generated a little different option value compared to any other methods. c.
The Effect of Maturity Date ( ) to Option Value The effect of maturity date changes to the double barrier knock-in option value estimator using the standard Monte Carlo method, antithetic variate, and control variate are presented in Figure 4. In Figure 4, it can be seen that the double barrier knock-in option value is getting bigger if the maturity date is also getting longer. The double barrier knock-in option values resulting from the standard Monte Carlo, antithetic variate, and control variate methods are relatively similar. The maturity date has a positive correlation with the option value. If the maturity date is getting longer, the option value is also getting bigger. d.
The Effect of Volatility ( ) to Option Value The effect of the volatility changes to the double barrier knock-in option value by standard Monte Carlo, antithetic variate, and control variate methods are presented in Figure 5. In Figure 5, it can be seen that the double barrier knock-in option value is getting bigger as the volatility is also getting bigger. The call option value is bigger than the put option value, with a volatility of 10%. However, the put option value is bigger than the call option value if the volatility is more than 10%. Volatility has a positive correlation with the option value. If the volatility is getting bigger, then the option value is getting bigger too.

CONCLUSIONS
In this paper, it is found that the Monte Carlo control variate method gave the value of the double barrier knock-in option faster than those of antithetic variate and standard Monte Carlo. The control variate method has the smallest relative error, followed by antithetic variate and standard Monte Carlo. The Standard Monte Carlo method has the shortest computation time, followed by control variate and antithetic variates. Compared to the other methods, the control variate methods are the most effective and efficient in determining the double barrier knock-in option value based on the option value, relative error, and computation time.
The increase in strike price makes the put double barrier knock-in option value to be higher and calls the double barrier knock-in option value to be lower. The increase in initial stock price makes the call double barrier knock-in option value to be higher and put double barrier knock-in option value to be lower. The increase in maturity date makes the double barrier knock-in option value to be higher. The increase in volatility makes the double barrier knock-in option to be higher.